Theorem xlt2add 9816

Description: Extended real version of lt2add 8343. Note that ltleadd 8344, which has weaker assumptions, is not true for the extended reals (since 0 + +oo < 1 + +oo fails). (Contributed by Mario Carneiro, 23-Aug-2015.)

Assertion

Ref	Expression
xlt2add	|- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A < C /\ B < D ) -> ( A +e B ) < ( C +e D ) ) )

Proof of Theorem xlt2add

Step	Hyp	Ref	Expression
1		xaddcl 9796	. . . . . . . 8 |- ( ( A e. RR* /\ B e. RR* ) -> ( A +e B ) e. RR* )
2	1	3ad2ant1 1008	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) e. RR* )
3	2	adantr 274	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) e. RR* )
4		simp1l 1011	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A e. RR* )
5		simp2r 1014	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D e. RR* )
6		xaddcl 9796	. . . . . . . 8 |- ( ( A e. RR* /\ D e. RR* ) -> ( A +e D ) e. RR* )
7	4, 5, 6	syl2anc 409	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e D ) e. RR* )
8	7	adantr 274	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e D ) e. RR* )
9		xaddcl 9796	. . . . . . . 8 |- ( ( C e. RR* /\ D e. RR* ) -> ( C +e D ) e. RR* )
10	9	3ad2ant2 1009	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e D ) e. RR* )
11	10	adantr 274	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( C +e D ) e. RR* )
12		simp3r 1016	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B < D )
13	12	adantr 274	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> B < D )
14		simp1r 1012	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B e. RR* )
15	14	adantr 274	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> B e. RR* )
16	5	adantr 274	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> D e. RR* )
17		simprl 521	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A e. RR )
18		xltadd2 9813	. . . . . . . 8 |- ( ( B e. RR* /\ D e. RR* /\ A e. RR ) -> ( B < D <-> ( A +e B ) < ( A +e D ) ) )
19	15, 16, 17, 18	syl3anc 1228	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( B < D <-> ( A +e B ) < ( A +e D ) ) )
20	13, 19	mpbid 146	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) < ( A +e D ) )
21		simp3l 1015	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A < C )
22	21	adantr 274	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A < C )
23	4	adantr 274	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> A e. RR* )
24		simp2l 1013	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C e. RR* )
25	24	adantr 274	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> C e. RR* )
26		simprr 522	. . . . . . . 8 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> D e. RR )
27		xltadd1 9812	. . . . . . . 8 |- ( ( A e. RR* /\ C e. RR* /\ D e. RR ) -> ( A < C <-> ( A +e D ) < ( C +e D ) ) )
28	23, 25, 26, 27	syl3anc 1228	. . . . . . 7 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A < C <-> ( A +e D ) < ( C +e D ) ) )
29	22, 28	mpbid 146	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e D ) < ( C +e D ) )
30	3, 8, 11, 20, 29	xrlttrd 9745	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ ( A e. RR /\ D e. RR ) ) -> ( A +e B ) < ( C +e D ) )
31	30	anassrs 398	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D e. RR ) -> ( A +e B ) < ( C +e D ) )
32		pnfxr 7951	. . . . . . . . . . . 12 |- +oo e. RR*
33	32	a1i 9	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> +oo e. RR* )
34		pnfge 9725	. . . . . . . . . . . 12 |- ( C e. RR* -> C <_ +oo )
35	24, 34	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C <_ +oo )
36	4, 24, 33, 21, 35	xrltletrd 9747	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A < +oo )
37		npnflt 9751	. . . . . . . . . . 11 |- ( A e. RR* -> ( A < +oo <-> A =/= +oo ) )
38	4, 37	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A < +oo <-> A =/= +oo ) )
39	36, 38	mpbid 146	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> A =/= +oo )
40		pnfge 9725	. . . . . . . . . . . 12 |- ( D e. RR* -> D <_ +oo )
41	5, 40	syl 14	. . . . . . . . . . 11 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D <_ +oo )
42	14, 5, 33, 12, 41	xrltletrd 9747	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B < +oo )
43		npnflt 9751	. . . . . . . . . . 11 |- ( B e. RR* -> ( B < +oo <-> B =/= +oo ) )
44	14, 43	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( B < +oo <-> B =/= +oo ) )
45	42, 44	mpbid 146	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> B =/= +oo )
46		xaddnepnf 9794	. . . . . . . . 9 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ ( B e. RR* /\ B =/= +oo ) ) -> ( A +e B ) =/= +oo )
47	4, 39, 14, 45, 46	syl22anc 1229	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) =/= +oo )
48		npnflt 9751	. . . . . . . . 9 |- ( ( A +e B ) e. RR* -> ( ( A +e B ) < +oo <-> ( A +e B ) =/= +oo ) )
49	2, 48	syl 14	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( ( A +e B ) < +oo <-> ( A +e B ) =/= +oo ) )
50	47, 49	mpbird 166	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) < +oo )
51	50	adantr 274	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( A +e B ) < +oo )
52		oveq2 5850	. . . . . . 7 |- ( D = +oo -> ( C +e D ) = ( C +e +oo ) )
53		mnfxr 7955	. . . . . . . . . . 11 |- -oo e. RR*
54	53	a1i 9	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo e. RR* )
55		mnfle 9728	. . . . . . . . . . 11 |- ( A e. RR* -> -oo <_ A )
56	4, 55	syl 14	. . . . . . . . . 10 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo <_ A )
57	54, 4, 24, 56, 21	xrlelttrd 9746	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < C )
58		nmnfgt 9754	. . . . . . . . . 10 |- ( C e. RR* -> ( -oo < C <-> C =/= -oo ) )
59	24, 58	syl 14	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < C <-> C =/= -oo ) )
60	57, 59	mpbid 146	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> C =/= -oo )
61		xaddpnf1 9782	. . . . . . . 8 |- ( ( C e. RR* /\ C =/= -oo ) -> ( C +e +oo ) = +oo )
62	24, 60, 61	syl2anc 409	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e +oo ) = +oo )
63	52, 62	sylan9eqr 2221	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( C +e D ) = +oo )
64	51, 63	breqtrrd 4010	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = +oo ) -> ( A +e B ) < ( C +e D ) )
65	64	adantlr 469	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D = +oo ) -> ( A +e B ) < ( C +e D ) )
66		simpr 109	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> D = -oo )
67		mnfle 9728	. . . . . . . . . 10 |- ( B e. RR* -> -oo <_ B )
68	14, 67	syl 14	. . . . . . . . 9 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo <_ B )
69	54, 14, 5, 68, 12	xrlelttrd 9746	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < D )
70		nmnfgt 9754	. . . . . . . . 9 |- ( D e. RR* -> ( -oo < D <-> D =/= -oo ) )
71	5, 70	syl 14	. . . . . . . 8 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < D <-> D =/= -oo ) )
72	69, 71	mpbid 146	. . . . . . 7 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> D =/= -oo )
73	72	adantr 274	. . . . . 6 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> D =/= -oo )
74	66, 73	pm2.21ddne 2419	. . . . 5 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ D = -oo ) -> ( A +e B ) < ( C +e D ) )
75	74	adantlr 469	. . . 4 |- ( ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) /\ D = -oo ) -> ( A +e B ) < ( C +e D ) )
76		elxr 9712	. . . . . 6 |- ( D e. RR* <-> ( D e. RR \/ D = +oo \/ D = -oo ) )
77	5, 76	sylib 121	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( D e. RR \/ D = +oo \/ D = -oo ) )
78	77	adantr 274	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) -> ( D e. RR \/ D = +oo \/ D = -oo ) )
79	31, 65, 75, 78	mpjao3dan 1297	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A e. RR ) -> ( A +e B ) < ( C +e D ) )
80		simpr 109	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> A = +oo )
81	39	adantr 274	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> A =/= +oo )
82	80, 81	pm2.21ddne 2419	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = +oo ) -> ( A +e B ) < ( C +e D ) )
83		oveq1 5849	. . . . 5 |- ( A = -oo -> ( A +e B ) = ( -oo +e B ) )
84		xaddmnf2 9785	. . . . . 6 |- ( ( B e. RR* /\ B =/= +oo ) -> ( -oo +e B ) = -oo )
85	14, 45, 84	syl2anc 409	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo +e B ) = -oo )
86	83, 85	sylan9eqr 2221	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> ( A +e B ) = -oo )
87		xaddnemnf 9793	. . . . . . 7 |- ( ( ( C e. RR* /\ C =/= -oo ) /\ ( D e. RR* /\ D =/= -oo ) ) -> ( C +e D ) =/= -oo )
88	24, 60, 5, 72, 87	syl22anc 1229	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( C +e D ) =/= -oo )
89		nmnfgt 9754	. . . . . . 7 |- ( ( C +e D ) e. RR* -> ( -oo < ( C +e D ) <-> ( C +e D ) =/= -oo ) )
90	10, 89	syl 14	. . . . . 6 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( -oo < ( C +e D ) <-> ( C +e D ) =/= -oo ) )
91	88, 90	mpbird 166	. . . . 5 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> -oo < ( C +e D ) )
92	91	adantr 274	. . . 4 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> -oo < ( C +e D ) )
93	86, 92	eqbrtrd 4004	. . 3 |- ( ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) /\ A = -oo ) -> ( A +e B ) < ( C +e D ) )
94		elxr 9712	. . . 4 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
95	4, 94	sylib 121	. . 3 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
96	79, 82, 93, 95	mpjao3dan 1297	. 2 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) /\ ( A < C /\ B < D ) ) -> ( A +e B ) < ( C +e D ) )
97	96	3expia 1195	1 |- ( ( ( A e. RR* /\ B e. RR* ) /\ ( C e. RR* /\ D e. RR* ) ) -> ( ( A < C /\ B < D ) -> ( A +e B ) < ( C +e D ) ) )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 967   /\ w3a 968   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982  (class class class)co 5842   RRcr 7752   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   < clt 7933   <_ cle 7934   +ecxad 9706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-1cn 7846  ax-1re 7847  ax-icn 7848  ax-addcl 7849  ax-addrcl 7850  ax-mulcl 7851  ax-addcom 7853  ax-addass 7855  ax-i2m1 7858  ax-0id 7861  ax-rnegex 7862  ax-pre-ltirr 7865  ax-pre-ltwlin 7866  ax-pre-lttrn 7867  ax-pre-ltadd 7869

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-if 3521  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-po 4274  df-iso 4275  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938  df-le 7939  df-xadd 9709

This theorem is referenced by:  bldisj  13051